Problem: Simplify and expand the following expression: $ \dfrac{x + 1}{x + 2}+\dfrac{3x + 4}{4x + 2} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(x + 2)(4x + 2)$ Multiply the first term by $\dfrac{4x + 2}{4x + 2}$ $ \begin{align*} \dfrac{x + 1}{x + 2} \times \dfrac{4x + 2}{4x + 2} & = \dfrac{(x + 1)(4x + 2)}{(x + 2)(4x + 2)} \\ & = \dfrac{4x^2 + 6x + 2}{(x + 2)(4x + 2)}\end{align*} $ Multiply the second term by $\dfrac{x + 2}{x + 2}$ $ \begin{align*} \dfrac{3x + 4}{4x + 2} \times \dfrac{x + 2}{x + 2} & = \dfrac{(3x + 4)(x + 2)}{(4x + 2)(x + 2)} \\ & = \dfrac{3x^2 + 10x + 8}{(4x + 2)(x + 2)}\end{align*} $ Now we have: $ = \dfrac{4x^2 + 6x + 2}{(x + 2)(4x + 2)} + \dfrac{3x^2 + 10x + 8}{(4x + 2)(x + 2)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{4x^2 + 6x + 2 + 3x^2 + 10x + 8}{(x + 2)(4x + 2)} $ $ = \dfrac{7x^2 + 16x + 10}{(x + 2)(4x + 2)}$ Expand the denominator: $ = \dfrac{7x^2 + 16x + 10}{4x^2 + 10x + 4}$